Energy Decay in the Dynamics of Complex Bodies with Spreading Microstructures Represented by $3D$ Vectors

Abstract

AbstractWe analyze complex bodies with active microstructure described by a vector field with values $\nu \in \mathbb{R}^{3}$ ν ∈ R 3 , which complements the displacement $u\in \mathbb{R}^{3}$ u ∈ R 3 . We consider linear elastic constitutive structures for macroscopic and microstructural stresses, diffusion of microstructures, and a viscoelastic-substrate-type effect, which has a regularizing influence that we eliminate at a later stage of our analysis. For weak solutions of the balance equations, we prove decay inequalities in $W^{1,2}$ W 1 , 2 and also for the $p$ p -power of the norms of $u$ u and $\nu $ ν in $L^{p}$ L p spaces under slip-without-friction-type boundary conditions.